Theorem REF  is a discretized projection theorem in the spirit of [1]}, [2]}. In [1]}, Bourgain proved that if \(X\subset [0,1]^2\)  is a set satisfying (REF ), and if \(\Theta \subset S^1\)  is a large (and non-concentrated) set of directions, then there must exist a direction \(e\in \Theta \)  so that the associated orthogonal projection \(\pi _{e}(q)\)  satisfies (REF ). The set \(\Theta \)  must be large, since if we define \(X=[0,1]^2 \cap (\delta ^{\alpha }\mathbb {Z})^2\) , then whenever \(s\)  and \(t\)  are integers with \(|s|\)  and \(|t|\)  small, the projection \((x,y)\cdot (s,t)\)  will have small cardinality and thus fail to satisfy (REF ). Thus it is new, and rather surprising, that Theorem REF  only requires three projections.
